Question

This is a network of streets We can observe the hourly flow of cars into this network’s entrances, and out of its exits. Page 5 East Winooski West Winooski Willow Jay Shelburne into 80 50 65 − 40 out of 30 5 70 55 75 (Note that to reach Jay a car must enter the network via some other road first, which is why there is no ‘into Jay’ entry in the table. Note also that over a long period of time, the total in must approximately equal the total out, which is why both rows add to 235 cars.) Once inside the network, the traffic may flow in different ways, perhaps filling Willow and leaving Jay mostly empty, or perhaps flowing in some other way. Kirchhoff’s Laws give the limits on that freedom. Suppose that someone proposes construction for Winooski Avenue East between Willow and Jay, and traffic on that block will be reduced. What is the least amount of traffic flow that can we can allow on that block without disrupting the hourly flow into and out of the network?

Answer 1

To determine the least amount of traffic flow that can be allowed on the block of Winooski Avenue East between Willow and Jay without disrupting the hourly flow into and out of the network, we can use Kirchhoff's Laws, which state that the total flow of traffic into a node equals the total flow of traffic out of the node, and the sum of the flows around a loop is zero.

Step-by-step explanation:

Let's define the flow of traffic on the block of Winooski Avenue East between Willow and Jay as x, and set up a system of equations based on Kirchhoff's Laws:

For the node at Willow:

65 + x = 70 + 55

For the node at East Winooski:

80 = 30 + x + 40

For the node at West Winooski:

50 = 5 + 75 - x

For the node at Shelburne:

x = 65 + 40 - 50

Simplifying each equation, we have:

x = 10 (from the Willow equation)

x = 10 (from the East Winooski equation)

x = 20 (from the West Winooski equation)

x = 55 (from the Shelburne equation)

The minimum value of x that satisfies all of these equations is 20. Therefore, the least amount of traffic flow that can be allowed on the block of Winooski Avenue East between Willow and Jay without disrupting the hourly flow into and out of the network is 20.